Figure REF (a) shows selected theoretical \(\pi \) -\(\sigma \) -polarized trRIXS spectra. The incident energy \({\omega _{\rm i}}=1.8 t_\mathrm {h}\)  (defined relative to the absorption edge) is determined according to the equilibrium resonance profile for the singly-occupied initial state. By varying the momentum transfer \(q\) , the equilibrium RIXS spectrum of a quarter-filled EHM displays a two-spinon continuum (\(t\!=\!-5t_h^{-1}\!<\!-\sigma _{\rm pump}\) ), as shown in Figs. REF . Different from undoped antiferromagnets [1]}, low-energy spin excitations of a doped extended Hubbard model mainly lie near the nesting vector \(q=2k_F\)  of spinon Fermi surface [2]}. At large momenta (\(q\ge \pi /2\) ), the trRIXS spectral weights are gradually suppressed by the x-ray scattering matrix elements [3]}, [4]}, which we divide out as shown in Eq. (REF ) when evaluating the spin QFIs. When evaluating the time-dependent QFI, we focus on a specific momentum, \(q = \pi /6\) , which exhibits the most evident spectral changes and captures the longest-range correlations supported by our system size.
